A205511 -id:A205511 - cp's OEIS frontend

A205533 Intersection of A205302 and A205511.

19, 43, 101, 197, 457, 461, 643, 743, 859, 883, 937, 1301, 1483, 1579, 1877, 1949, 1997, 2083, 2129, 2141, 2221, 2237, 2251, 2381, 2539, 2609, 2617, 2659, 2663, 3019, 3221, 3389, 3461, 3701, 4003, 4157, 4517, 4637, 5573, 5741, 5783, 6763, 6899, 6907, 7349, 7757, 7877

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Jan 28 2012

Both Hamming's distances between a(n) and the previous prime and between a(n) and the next prime equal 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A205302, A205511, A205510, A205509, A001097, A001359, A006512.

Select[Partition[Prime[Range[1000]], 3, 1], DigitCount[BitXor[#[[1]], #[[2]]], 2, 1] == DigitCount[BitXor[#[[2]], #[[3]]], 2, 1] == 1 &] [[;; , 2]] (* Amiram Eldar, Aug 06 2023 *)

More terms from Amiram Eldar, Aug 06 2023

A205510 Binary Hamming distance between prime(n) and prime(n+1).

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 4, 2, 1, 1, 3, 3, 2, 6, 1, 3, 2, 3, 2, 3, 1, 1, 2, 2, 3, 3, 6, 2, 1, 4, 1, 2, 5, 1, 2, 4, 2, 2, 6, 1, 1, 2, 2, 4, 2, 2, 2, 4, 2, 7, 2, 2, 1, 3, 2, 1, 5, 3, 1, 3, 1, 5, 3, 2, 2, 4, 2, 1, 3, 3, 1, 6, 1, 3, 1, 4, 2, 2, 4, 2, 2, 5, 1, 1, 1, 3, 2, 3, 2, 2, 1, 2, 7, 1, 3, 5

Offset: 1

Vladimir Shevelev, Jan 28 2012

We call "Hamming's twin primes" the pairs of consecutive primes (p,q) with Hamming distance 1. They are (2,3), (5,7), (17,19,), (19,23), (29,31), (41,43), (43,47), (67,71), (97,101), ..., (A205511,A205302). As in Twin Primes Conjecture, we conjecture that there exist infinitely many Hamming's twin pairs.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A205511, A205302, A205509, A001511, A345985.

a:= n-> add(i, i=Bits[GetBits](Bits[Xor](ithprime(n), ithprime(n+1)), 0..-1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 11 2017

Table[Count[IntegerDigits[BitXor[Prime[n],Prime[n+1]],2],1],{n,100}] (* Jayanta Basu, May 26 2013 *)

A205510(n)=norml2(binary(bitxor(prime(n),prime(n+1))))  \\ M. F. Hasler, Jan 29 2012

a(n,p=prime(n),q=nextprime(p+1))=hammingweight(bitxor(p,q)) \\ Charles R Greathouse IV, Nov 15 2022

Corrected a(24) and a(25) by M. F. Hasler, Jan 29 2012

Added "binary" to definition. - N. J. A. Sloane, Jul 09 2021

A206853 a(1)=1, for n>1, a(n) is the least number > a(n-1) such that the Hamming distance D(a(n-1), a(n)) = 2.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 14, 22, 26, 28, 31, 47, 55, 59, 61, 62, 94, 110, 118, 122, 124, 127, 191, 223, 239, 247, 251, 253, 254, 382, 446, 478, 494, 502, 506, 508, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1534, 1790, 1918, 1982, 2014, 2030, 2038, 2042

Offset: 1

Vladimir Shevelev, Feb 13 2012

For integers a, b, denote by a<+>b the least c>=a, such that D(a,c)=b (note that, generally speaking, a<+>b differs from b<+>a). Then a(n+1)=a(n)<+>2. Thus this sequence is a Hamming analog of odd numbers 1,3,5,...
A Hamming analog of nonnegative integers is A000225 and a Hamming analog of the triangular numbers is A000975.
All terms are odious (A000069).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A182187 (n<+>2), A207063 (starting from 0).

Cf. A000225, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A000069.

read("transforms");
Hamming := proc(a,b)
        XORnos(a,b) ;
        wt(%) ;
end proc:
Dplus := proc(a,b)
        for c from a to 1000000 do
                if Hamming(a,c)=b then
                        return c;
                end if;
        end do:
        return -1 ;
end proc:
A206853 := proc(n)
        option remember;
        if n = 1 then
                1;
        else
                Dplus(procname(n-1),2) ;
        end if;
end proc: # R. J. Mathar, Apr 05 2012

myHammingDistance[n_, m_] := Module[{g = Max[m, n], h = Min[m, n]}, b1 = IntegerDigits[g, 2]; b2 = IntegerDigits[h, 2, Length[b1]]; HammingDistance[b1, b2]]; t = {1}; Do[If[myHammingDistance[t[[-1]], n] == 2, AppendTo[t, n]], {n, 2, 2042}]; t (* T. D. Noe, Mar 07 2012 *)
t={x=1}; Do[i=x+1; While[Count[IntegerDigits[BitXor[x,i],2],1]!=2,i++]; AppendTo[t,x=i],{n,53}]; t (* Jayanta Basu, May 26 2013 *)

next_A206853(n)={my(b=binary(n)); until(norml2(binary(n)-b)==2, n++>=2^#b & b=concat(0,b)); n}
print1(n=1); for(i=1,99,print1(","n=next_A206853(n)))  \\ M. F. Hasler, Apr 07 2012

A205302 Greater of Hamming's twin primes.

Original entry on oeis.org

3, 7, 19, 23, 31, 43, 47, 71, 101, 103, 139, 151, 167, 197, 199, 271, 283, 311, 317, 367, 383, 397, 409, 457, 461, 463, 503, 523, 571, 619, 643, 647, 677, 743, 751, 773, 811, 823, 859, 863, 883, 887, 911, 937, 941, 991, 1013, 1051, 1063, 1117, 1231, 1279, 1291, 1301, 1303, 1483, 1487

Offset: 1

Vladimir Shevelev, Jan 28 2012

The Hamming distance between a(n) and the previous prime is 1 (cf. A205510).

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A205510, A205511, A205533.

Cf. A001097, A001359, A006512.

Select[Partition[Prime[Range[250]], 2, 1], DigitCount[BitXor[First[#], Last[#]], 2, 1] == 1 &] [[;; , 2]] (* Amiram Eldar, Aug 06 2023 *)

n=0;for(i=1,80,until(A205510(n++)==1,);print1(prime(n+1)","))  \\ M. F. Hasler, Jan 29 2012

a(10)-a(57) from Peter J. C. Moses, Jan 28 2012

Values verified by M. F. Hasler, Jan 29 2012

A205649 Hamming distance between twin primes.

Original entry on oeis.org

2, 1, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 6, 1, 2, 4, 1, 1, 3, 2, 2, 4, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 7, 1, 1, 1, 1, 3, 2, 2, 1, 4, 3, 2, 2, 1, 1, 2, 4, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 1, 2, 1, 2, 1, 1, 5, 1, 7, 3, 1, 1, 1, 1, 3, 4, 5, 2, 1, 2

Offset: 1

Vladimir Shevelev, Jan 30 2012

Twin primes for which a(n)=1 are in A122565.

Conjecture: The sequence is unbounded.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A205510, A001097, A001359, A006512, A205302, A205511, A205533.

nn = 1000; ps = Prime[Range[nn]]; t = {}; Do[If[ps[[n]] + 2 == ps[[n + 1]], AppendTo[t, ps[[n]]]], {n, nn - 1}]; Table[b2 = IntegerDigits[t[[k]] + 2, 2];  b1 = IntegerDigits[t[[k]], 2, Length[b2]]; Total[Abs[b1 - b2]], {k, Length[t]}] (* T. D. Noe, Jan 30 2012 *)

A001359(n) == -1 (mod 2^a(n)).

A206960 a(0)=0, a(1)=1, for n>1, a(n) = n<>n, where k<>m = k<+>k<+>...<+>k is the m-1-fold iteration of the operation <+> defined in A206853.

Original entry on oeis.org

0, 1, 4, 9, 31, 66, 190, 385, 1022, 2055, 5112, 10242, 24572, 49153, 114686, 229377, 524287, 1048590, 2359281, 4718599, 10485753, 20971521, 46137342, 92274689, 201326590, 402653191, 872415224, 1744830466, 3758096380, 7516192769, 16106127358, 32212254721

Offset: 0

Vladimir Shevelev and Konstantin Shukhmin (shukhmin(AT)callsouth.net.nz), Feb 14 2012

A Hamming's analog of n^2.

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853.

f[n_,k_]:=Module[{i=n+1},While[Count[IntegerDigits[BitXor[n,i],2],1]!=k,i++]; i]; Join[{0,1},Table[Nest[f[#,k]&,k,k-1],{k,2,18}]] (* Jayanta Basu, May 26 2013 *)

A010060(a(n)) = A010060(n). - Vladimir Shevelev and Alois P. Heinz, Feb 17 2012

a(11)-a(31) from Alois P. Heinz, Feb 16 2012

A182187 a(n) is the least m >= n such that the Hamming distance D(n,m) = 2.

Original entry on oeis.org

3, 2, 4, 5, 7, 6, 10, 11, 11, 10, 12, 13, 15, 14, 22, 23, 19, 18, 20, 21, 23, 22, 26, 27, 27, 26, 28, 29, 31, 30, 46, 47, 35, 34, 36, 37, 39, 38, 42, 43, 43, 42, 44, 45, 47, 46, 54, 55, 51, 50, 52, 53, 55, 54, 58, 59, 59, 58, 60, 61, 63, 62, 94, 95, 67, 66, 68

Offset: 0

Vladimir Shevelev, Apr 17 2012

a(n) = n<+>2 (see comment in A206853).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A206853 (trajectory of 1), A207063 (trajectory of 0).
Cf. A209544 (primes which are not terms), A209554 (and also not n<+>3).
Cf. A086799 ((n-1)<+>1), A182209 (n<+>3), A182336 (n<+>4).
Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206960, A007814.

HD:= (i, j)-> add(h, h=Bits[Split](Bits[Xor](i, j))):
a:= proc(n) local c;
      for c from n do if HD(n, c)=2 then return c fi od
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 17 2012

t={}; Do[i=n+1; While[Count[IntegerDigits[BitXor[n,i],2],1]!=2,i++]; AppendTo[t,i],{n,0,66}]; t (* Jayanta Basu, May 26 2013 *)

a(n) = bitxor(n, 3<>1+1,2)); \\ Kevin Ryde, Jul 09 2021

def a(n):
  m = n + 1
  while bin(n^m).count('1') != 2: m += 1
  return m
print([a(n) for n in range(67)]) # Michael S. Branicky, Mar 02 2021

def A182187(n):
    S = n.bits(); T = S; c = n; L = len(S)
    while true:
         H = sum(a != b for a, b in zip(S, T))
         if H == 2: return c
         c += 1; T = c.bits()
         if len(T) > L: L += 1; S.append(0)
[A182187(n) for n in (0..66)]   # Peter Luschny, May 26 2013

If n is odd, then a(n) = n+2^(A007814(n+1)-1); if n == 2 (mod 4), then a(n) = n+2^(A007814(n+2)-1); if n == 0 (mod 4), then a(n) = n+3.
Using this formula, we can prove the conjecture formulated in comment in A209554 in the case k=2. Moreover, let us show that if N does not have the form 8*t or 8*t+1, then it can be represented in the form n<+>2. Indeed, in the cases N = 8*t+2, 8*t+4, 8*t+6, 8*t+3, 8*t+5 and 8*t+7 it is sufficient to choose n=N-4, n=N-2, n=N-1, n=N-3, n=N-2 and n = N-3 respectively; in the cases 8*t, 8*t+1, for every choice of n <= N, we do not obtain the equality n<+>2 = N.
In addition, note that n<+>1 = n + 2^A007814(n+1) = A086799(n+1).

More terms from Alois P. Heinz, Apr 17 2012

A207017 Numbers m for which there exists a number 1k, where operation <+> is defined in A206853.

Original entry on oeis.org

4, 7, 9, 11, 13, 14, 16, 18, 21, 22, 24, 26, 28, 31, 33, 35, 39, 41, 44, 46, 47, 49, 50, 53, 55, 56, 57, 59, 61, 62, 63, 66, 70, 73, 79, 82, 83, 84, 89, 93, 94, 96, 97, 102, 104, 110, 111, 112, 115, 116, 118, 120, 121, 122, 124, 125, 126, 127, 129, 131

Offset: 1

Vladimir Shevelev, Feb 14 2012

It is natural to call terms of the sequence "Hamming composite numbers" and to say that m is "H-divisible" by k.

			127 = b_21 for k=2, b_16 for k=4 and b_8 for k=5. Thus 127 is H-divisible by 2, 4 and 5 (and only by them).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000225, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206853, A207016.

A182209 a(n) is the least m >= n, such that the Hamming distance D(n,m) = 3.

Original entry on oeis.org

7, 6, 5, 4, 9, 8, 8, 9, 15, 14, 13, 12, 16, 17, 18, 19, 23, 22, 21, 20, 25, 24, 24, 25, 31, 30, 29, 28, 36, 37, 38, 39, 39, 38, 37, 36, 41, 40, 40, 41, 47, 46, 45, 44, 48, 49, 50, 51, 55, 54, 53, 52, 57, 56, 56, 57, 63, 62, 61, 60, 76, 77, 78, 79, 71, 70, 69

Offset: 0

Vladimir Shevelev, Apr 18 2012

a(n) = n<+>3 (see comment in A206853).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A086799 ((n-1)<+>1), A182187 (n<+>2), A182336 (n<+>4).
Cf. A209554 (primes which are not terms nor n<+>2 terms).
Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206960, A007814.

HD:= (i, j)-> add(h, h=Bits[Split](Bits[Xor](i, j))):
a:= proc(n) local c;
      for c from n do if HD(n, c)=3 then return c fi od
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 18 2012

a(n) = bitxor(n, if(bitand(n,14)==4, 13, 7<>2+1,2))); \\ Kevin Ryde, Jul 10 2021

def d(n, m): return bin(n^m).count('1')
def a(n):
    m = n+1
    while d(n, m) != 3: m += 1
    return m
print([a(n) for n in range(67)]) # Michael S. Branicky, Jul 06 2021

If n==i mod 8, then a(n) = n-2*i+7, i=0,1,2,3; if n==4 mod 16, then a(n) = n+5; if n==12 mod 16, then a(n) = n+2^(A007814(n+4)-2); if n==5 mod 16, then a(n) = n+3; if n==13 mod 16, then a(n) = n+2^(A007814(n+3)-2); if n==6 mod 8, then a(n) = n+2^(A007814(n+2)-2); if n==7 mod 8, then a(n) = n+2^(A007814(n+1)-2).

Using this formula, we can prove conjecture formulated in comment in A209554 in case k=3. Moreover, one can prove that N could be represented in form n<+>2 or n<+>3 iff N is not a number of the forms 32*t, 32*t+1. - Vladimir Shevelev, Apr 25 2012

A207016 a(1) = 3 , for n > 1 a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 3.

Original entry on oeis.org

3, 4, 9, 14, 18, 21, 24, 31, 39, 41, 46, 50, 53, 56, 63, 79, 83, 84, 89, 94, 102, 104, 111, 115, 116, 121, 126, 158, 166, 168, 175, 179, 180, 185, 190, 206, 210, 213, 216, 223, 231, 233, 238, 242, 245, 248, 255, 319, 335, 339, 340, 345, 350, 358, 360, 367, 371

Offset: 1

Vladimir Shevelev, Feb 14 2012

Odious and evil terms are alternating (cf. A000069, A001969).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A000069, A000225, A001969, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853.

a[1] = 3; a[n_] := a[n] = Module[{k = a[n - 1], m = a[n-1] + 1}, While[DigitCount[BitXor[k, m], 2, 1] != 3, m++]; m]; Array[a, 100] (* Amiram Eldar, Aug 06 2023 *)

list(n)=my(v=vector(n));v[1]=3;for(i=2,n,v[i]=v[i-1]; while(hammingweight(bitxor(v[i-1],v[i]++))!=3,));v \\ Charles R Greathouse IV, Mar 26 2013

cp's OEIS Frontend

A205533 Intersection of A205302 and A205511.

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A205510 Binary Hamming distance between prime(n) and prime(n+1).

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A206853 a(1)=1, for n>1, a(n) is the least number > a(n-1) such that the Hamming distance D(a(n-1), a(n)) = 2.

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A205302 Greater of Hamming's twin primes.

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A205649 Hamming distance between twin primes.

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A206960 a(0)=0, a(1)=1, for n>1, a(n) = n<*>n, where k<*>m = k<+>k<+>...<+>k is the m-1-fold iteration of the operation <+> defined in A206853.

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A182187 a(n) is the least m >= n such that the Hamming distance D(n,m) = 2.

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A206960 a(0)=0, a(1)=1, for n>1, a(n) = n<>n, where k<>m = k<+>k<+>...<+>k is the m-1-fold iteration of the operation <+> defined in A206853.